Figure 1.
The curves on the subgraphs (a) and (b) are the density functions of x(t) and y(t) in k = 1, k = 2, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
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On the limit quasi-shadowing property
May
2017, 37(5): 2881-2897.
doi: 10.3934/dcds.2017124
Persistence and stationary distribution of a stochastic predator-prey model under regime switching
| 1. | College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China |
| 2. | School of Science, China University of Petroleum (East China), Qingdao 266580, China |
| 3. | Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia |
| 4. | School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland |
Taking both white noise and colored environment noise into account, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation figures are presented to support the analytical findings.
Keywords:
Stochastic predator-prey model,
threshold,
Markov switching,
stationary distribution,
extinction.
Mathematics Subject Classification:
Primary: 34F05, 92D25; Secondary: 92Bxx.
Citation:
Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete and Continuous Dynamical Systems,
2017, 37
(5)
: 2881-2897.
doi: 10.3934/dcds.2017124
![]()
References:
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G. Q. Cai and Y. K. Lin,
Stochastic analysis of predator-prey type ecosystems, Ecol. Complex., 4 (2007), 242-249.
doi: 10.1016/j.ecocom.2007.06.011. |
| [2] |
L. S. Chen and Z. J. Jing,
The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Sci. Bull., 9 (1984), 521-523.
|
| [3] |
H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma,
A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.
doi: 10.1016/j.tpb.2004.06.010. |
| [4] |
C. Y. Ji, D. Q. Jiang and N. Z. Shi,
A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.
doi: 10.1016/j.jmaa.2010.11.008. |
| [5] |
R. Z. Khasminskii, C. Zhu and G. Yin,
Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051.
doi: 10.1016/j.spa.2006.12.001. |
| [6] |
H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages.
doi: 10.1155/2014/720283. |
| [7] |
X. Y. Li, D. Q. Jiang and X. R. Mao,
Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.
doi: 10.1016/j.cam.2009.06.021. |
| [8] |
X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao,
Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.
doi: 10.1016/j.jmaa.2010.10.053. |
| [9] |
X. Y. Li and X. R. Mao,
A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334.
doi: 10.1016/j.automatica.2012.06.045. |
| [10] |
H. Liu, Q. S. Yang and D. Q. Jiang,
The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825.
doi: 10.1016/j.automatica.2012.02.010. |
| [11] |
H. Liu, X. X. Li and Q. S. Yang,
The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.
doi: 10.1016/j.sysconle.2013.06.002. |
| [12] |
M. Liu and K. Wang,
Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418.
doi: 10.1016/j.mcm.2011.08.019. |
| [13] |
M. Liu and K. Wang,
Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154.
doi: 10.1016/j.mcm.2011.05.023. |
| [14] |
X. R. Mao, S. Sabanis and E. Renshaw,
Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
| [15] |
X. R. Mao,
Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
| [16] |
X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006. |
| [17] |
R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982. |
| [18] |
E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969. |
| [19] |
A. Settati and A. Lahrouz,
Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.
doi: 10.1016/j.amc.2014.07.012. |
| [20] |
Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa,
Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.
doi: 10.1016/j.jmaa.2005.11.009. |
| [21] |
D. Y. Xu, B. Li, S. J. Long and L. Y. Teng,
Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143.
doi: 10.1016/j.na.2014.05.004. |
| [22] |
D. Y. Xu, X. H. Wang and Z. G. Yang,
Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180.
doi: 10.1007/s11425-012-4553-1. |
| [23] |
T. Zhao, Y. Kung and H. L. Smith,
Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394.
doi: 10.1016/0362-546X(95)00230-S. |
| [24] |
C. Zhu and G. Yin,
Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
| [25] |
L. Zu, D. Q. Jiang and D. O'Regan,
Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.
doi: 10.1016/j.cnsns.2015.04.008. |
show all references
References:
| [1] |
G. Q. Cai and Y. K. Lin,
Stochastic analysis of predator-prey type ecosystems, Ecol. Complex., 4 (2007), 242-249.
doi: 10.1016/j.ecocom.2007.06.011. |
| [2] |
L. S. Chen and Z. J. Jing,
The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Sci. Bull., 9 (1984), 521-523.
|
| [3] |
H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma,
A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.
doi: 10.1016/j.tpb.2004.06.010. |
| [4] |
C. Y. Ji, D. Q. Jiang and N. Z. Shi,
A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.
doi: 10.1016/j.jmaa.2010.11.008. |
| [5] |
R. Z. Khasminskii, C. Zhu and G. Yin,
Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051.
doi: 10.1016/j.spa.2006.12.001. |
| [6] |
H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages.
doi: 10.1155/2014/720283. |
| [7] |
X. Y. Li, D. Q. Jiang and X. R. Mao,
Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.
doi: 10.1016/j.cam.2009.06.021. |
| [8] |
X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao,
Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.
doi: 10.1016/j.jmaa.2010.10.053. |
| [9] |
X. Y. Li and X. R. Mao,
A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334.
doi: 10.1016/j.automatica.2012.06.045. |
| [10] |
H. Liu, Q. S. Yang and D. Q. Jiang,
The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825.
doi: 10.1016/j.automatica.2012.02.010. |
| [11] |
H. Liu, X. X. Li and Q. S. Yang,
The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.
doi: 10.1016/j.sysconle.2013.06.002. |
| [12] |
M. Liu and K. Wang,
Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418.
doi: 10.1016/j.mcm.2011.08.019. |
| [13] |
M. Liu and K. Wang,
Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154.
doi: 10.1016/j.mcm.2011.05.023. |
| [14] |
X. R. Mao, S. Sabanis and E. Renshaw,
Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.
doi: 10.1016/S0022-247X(03)00539-0. |
| [15] |
X. R. Mao,
Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405.
doi: 10.1016/j.sysconle.2011.02.013. |
| [16] |
X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006. |
| [17] |
R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982. |
| [18] |
E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969. |
| [19] |
A. Settati and A. Lahrouz,
Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.
doi: 10.1016/j.amc.2014.07.012. |
| [20] |
Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa,
Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.
doi: 10.1016/j.jmaa.2005.11.009. |
| [21] |
D. Y. Xu, B. Li, S. J. Long and L. Y. Teng,
Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143.
doi: 10.1016/j.na.2014.05.004. |
| [22] |
D. Y. Xu, X. H. Wang and Z. G. Yang,
Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180.
doi: 10.1007/s11425-012-4553-1. |
| [23] |
T. Zhao, Y. Kung and H. L. Smith,
Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394.
doi: 10.1016/0362-546X(95)00230-S. |
| [24] |
C. Zhu and G. Yin,
Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
| [25] |
L. Zu, D. Q. Jiang and D. O'Regan,
Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.
doi: 10.1016/j.cnsns.2015.04.008. |
Figure 2.
The subgraphs (a) and (b) denote sample phase portrait of stochastic system and the corresponding deterministic system. The red, blue and black area denote x(t), y(t) in k = 1, k = 2 and converting between the above two states, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
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